概念：
贝叶斯定理：贝叶斯理论是以18世纪的一位神学家托马斯.贝叶斯(Thomas Bayes)命名。通常，事件A在事件B（发生）的条件下的概率，与事件B在事件A（发生）的条件下的概率是不一样的；然而，这两者是有确定的关系的，贝叶斯定理就是这种关系的陈述
朴素贝叶斯：朴素贝叶斯方法是基于贝叶斯定理和特征条件独立假设的分类方法。对于给定的训练数据集，首先基于特征条件独立假设学习输入/输出的联合概率分布；然后基于此模型，对给定的输入x，利用贝叶斯定理求出后验概率(Maximum A Posteriori)最大的输出y。
通俗的来讲，在给定数据集的前提下，对于一个新样本（未分类），在数据集中找到和新样本特征相同的样本，最后根据这些样本算出每个类的概率，概率最高的类即为新样本的类。
运算公式：
P( h | d) = P ( d | h ) * P( h) / P(d)
这里：
P ( h | d )：是因子h基于数据d的假设概率,叫做后验概率
P ( d | h ) : 是假设h为真条件下的数据d的概率
P( h) : 是假设条件h为真的时候的概率（和数据无关），它叫做h的先验概率
P(d) : 数据d的概率，和先验条件无关．
算法实现分解：
１ 数据处理：加载数据并把他们分成训练数据和测试数据
２ 汇总数据：汇总训练数据的概率以便后续计算概率和做预测
３ 结果预测： 通过给定的测试数据和汇总的训练数据做预测
４ 评估准确性：使用测试数据来评估预测的准确性
代码实现：
# Example of Naive Bayes implemented from Scratch in Python
import csv
import random
import math def loadCsv(filename):
lines = csv.reader(open(filename, "rb"))
dataset = list(lines)
for i in range(len(dataset)):
dataset[i] = [float(x) for x in dataset[i]]
return dataset def splitDataset(dataset, splitRatio):
trainSize = int(len(dataset) * splitRatio)
trainSet = []
copy = list(dataset)
while len(trainSet) < trainSize:
index = random.randrange(len(copy))
trainSet.append(copy.pop(index))
return [trainSet, copy] def separateByClass(dataset):
separated = {}
for i in range(len(dataset)):
vector = dataset[i]
if (vector[-1] not in separated):
separated[vector[-1]] = []
separated[vector[-1]].append(vector)
return separated def mean(numbers):
return sum(numbers)/float(len(numbers)) def stdev(numbers):
avg = mean(numbers)
variance = sum([pow(x-avg,2) for x in numbers])/float(len(numbers)-1)
return math.sqrt(variance) def summarize(dataset):
summaries = [(mean(attribute), stdev(attribute)) for attribute in zip(*dataset)]
del summaries[-1]
return summaries def summarizeByClass(dataset):
separated = separateByClass(dataset)
summaries = {}
for classValue, instances in separated.iteritems():
summaries[classValue] = summarize(instances)
return summaries def calculateProbability(x, mean, stdev):
exponent = math.exp(-(math.pow(x-mean,2)/(2*math.pow(stdev,2))))
return (1 / (math.sqrt(2*math.pi) * stdev)) * exponent def calculateClassProbabilities(summaries, inputVector):
probabilities = {}
for classValue, classSummaries in summaries.iteritems():
probabilities[classValue] = 1
for i in range(len(classSummaries)):
mean, stdev = classSummaries[i]
x = inputVector[i]
probabilities[classValue] *= calculateProbability(x, mean, stdev)
return probabilities def predict(summaries, inputVector):
probabilities = calculateClassProbabilities(summaries, inputVector)
bestLabel, bestProb = None, -1
for classValue, probability in probabilities.iteritems():
if bestLabel is None or probability > bestProb:
bestProb = probability
bestLabel = classValue
return bestLabel def getPredictions(summaries, testSet):
predictions = []
for i in range(len(testSet)):
result = predict(summaries, testSet[i])
predictions.append(result)
return predictions def getAccuracy(testSet, predictions):
correct = 0
for i in range(len(testSet)):
if testSet[i][-1] == predictions[i]:
correct += 1
return (correct/float(len(testSet))) * 100.0 def main():
filename = 'pima-indians-diabetes.data.csv'
splitRatio = 0.67
dataset = loadCsv(filename)
trainingSet, testSet = splitDataset(dataset, splitRatio)
print('Split {0} rows into train={1} and test={2} rows').format(len(dataset), len(trainingSet), len(testSet))
# prepare model
summaries = summarizeByClass(trainingSet)
# test model
predictions = getPredictions(summaries, testSet)
accuracy = getAccuracy(testSet, predictions)
print('Accuracy: {0}%').format(accuracy) main()
pima-indians-diabetes.data.csv的下载地址：
https://raw.githubusercontent.com/jbrownlee/Datasets/master/pima-indians-diabetes.data.csv
参考文档：
１ https://en.wikipedia.org/wiki/Naive_Bayes_classifier
２ https://machinelearningmastery.com/naive-bayes-classifier-scratch-python/
３ https://machinelearningmastery.com/naive-bayes-for-machine-learning/